1.
Suppose that \(f\) has a zero of multiplicity \(m\) at \(a\text{.}\) Explain why \(\frac 1 f\) has a pole of order \(m\) at \(a\text{.}\)
Exercises 9.4 Exercises
2.
Find the poles or removable singularities of the following functions and determine their orders:
(a)
\((z^2+1)^{-3}(z-1)^{-4}\)
(b)
\(z\cot(z)\)
(c)
\(z^{-5}\sin(z)\)
(d)
\(\ds \frac1{1-\exp(z)}\)
(e)
\(\ds \frac z{1-\exp(z)}\)
3.
Suppose \(f\) is a nonconstant entire function. Prove that any complex number is arbitrarily close to a number in \(f(\C)\text{.}\)
Hint.
4.
Evaluate the following integrals for \(\gg = C[0,3]\text{.}\)
(a)
\(\displaystyle \i \cot (z) \, \diff{z}\)
(b)
\(\displaystyle \i z^3 \cos ( \tfrac 3 z )\, \diff{z}\)
(c)
\(\displaystyle \i \frac{ \diff{z} }{ (z+4) ( z^2 + 1 ) }\)
(d)
\(\displaystyle \i z^2 \exp ( \tfrac 1 z )\, \diff{z}\)
(e)
\(\displaystyle \i \frac{ \exp (z) }{ \sinh (z) } \, \diff{z}\)
(f)
\(\displaystyle \i \frac{ i^{z+4} }{ (z^2 + 16)^2 } \, \diff{z}\)
5.
Suppose \(f\) has a simple pole (i.e., a pole of order 1) at \(z_0\) and \(g\) is holomorphic at \(z_0\text{.}\) Prove that
\begin{equation*}
\Res_{z=z_0} \bigl( f(z) \, g(z) \bigr) \ = \ g(z_0) \,
\Res_{z=z_0} \bigl( f(z) \bigr) \, \text{.}
\end{equation*}
6.
Find the residue of each function at \(0\text{:}\)
(a)
\(z^{-3}\cos(z)\)
(b)
\(\csc(z)\)
(c)
\(\frac{z^2+4z+5}{z^2+z}\)
(d)
\(\exp(1-\tfrac1z)\)
(e)
\(\frac{\exp(4z)-1}{\sin^2(z)}\)
7.
Use residues to evaluate the following integrals:
(a)
\(\ds \int_{ C[i-1,1] }\frac{\diff{z}}{z^4+4}\)
(b)
\(\ds \int_{ C[i,2] }\frac{\diff{z}}{z(z^2+z-2)}\)
(c)
\(\ds \int_{ C[0,2] }\frac{\exp(z)}{z^3+z}\diff{z}\)
(d)
\(\ds \int_{ C[0,1] }\frac{\diff{z}}{z^2\sin z}\)
(e)
\(\ds \int_{ C[0,3] }\frac{\exp(z)}{(z+2)^2 \sin z}
\diff{z}\)
(f)
\(\ds \int_{ C[\pi,1] } \frac{ \exp(z) }{ \sin(z) \cos(z) }
\diff{z}\)
8.
Use the Residue Theorem 9.2.2 to re-prove Cauchy’s Integral Formulas (Theorem 4.4.5, Theorem 5.1.1 and Corollary 8.1.12.
9.
10.
(a)
(b)
Find \(\Res_{z=z_0} (f')\text{.}\)
11.
Extend Proposition 9.2.7 by proving, if \(f\) and \(g\) are holomorphic at \(z_0\text{,}\) which is a double zero of \(g\text{,}\) then
\begin{equation*}
\Res_{z=z_0} \left( \frac{ f(z) }{ g(z) } \right) \ = \ \frac{
6 \, f'(z_0) \, g''(z_0) - 2 \, f(z_0) \, g'''(z_0) }{ 3 \,
g''(z_0)^2 } \, \text{.}
\end{equation*}
12.
Compute \(\ds \int_{ C[2,3] } \frac{ \cos(z) }{ \sin^2(z) }
\,\diff{z} \text{.}\)
13.
Generalize Example 5.3.5 and Exercise 5.4.18 as follows: Let \(p(x)\) and \(q(x)\) be polynomials such that \(q(x) \ne 0\) for \(x \in \R\) and the degree of \(q(x)\) is at least two larger than the degree of \(p(x)\text{.}\) Prove that \(\int_{ -\infty }^\infty \frac{ p(x) }{ q(x) } \,
\diff{x}\) equals \(2 \pi i\) times the sum of the residues of \(\frac{ p(z) }{ q(z) }\) at all poles in the upper half plane.
14.
Compute \(\ds \int_{-\infty}^\infty \frac{ \diff{x} }{
(1+x^2)^2 } \text{.}\)
15.
Generalize Exercise 5.4.19 by deriving conditions under which we can compute \(\int_{ -\infty }^\infty \frac{ p(x) \cos(x) }{ q(x) } \,
\diff{x}\) for polynomials \(p(x)\) and \(q(x)\text{,}\) and give a formula for this integral along the lines of Exercise 9.4.13.
16.
Compute \(\ds \int_{-\infty}^\infty \frac{ \sin(x) }{ 1+x^4 }
\, \diff{x} \text{.}\)
17.
Suppose \(f\) is entire and \(a,
b \in \C\) with \(a \ne b\) and \(|a|, |b| \lt R\text{.}\) Evaluate
\begin{equation*}
\int_{ C[0,R] } \frac{ f(z) }{ (z-a)(z-b) } \, \diff{z}
\end{equation*}
and use this to give an alternate proof of Liouville’s Corollary 5.3.4.
18.
19.
Suppose \(f\) is meromorphic in the region \(G\text{,}\) \(g\) is holomorphic in \(G\text{,}\) and \(\gg\) is a positively oriented, simple, closed, piecewise smooth path that does not pass through any zero or pole of \(f\text{,}\) and \(\gg \sim_G 0\text{.}\) Denote the zeros and poles of \(f\) inside \(\gg\) by \(z_1, \dots,
z_j\) and \(p_1, \dots,
p_k\text{,}\) respectively, counted according to multiplicity. Prove that
\begin{equation*}
\frac{1}{2 \pi i} \i g \, \frac{f'}{f} \ = \ \sum_{m=1}^j
g(z_m) - \sum_{n=1}^k g(p_n) \, \text{.}
\end{equation*}
20.
Find the number of zeros of
(a)
(b)
(c)
21.
Give another proof of the Fundamental Theorem of Algebra (Theorem 5.3.2), using Rouché’s Theorem 9.3.3.
Hint.
If \(p(z) = a_n
z^n + a_{n-1} z^{n-1} + \dots + a_1 z + 1\text{,}\) let \(f(z) = a_n z^n\) and \(g(z) = a_{n-1} z^{n-1} +
a_{n-2} z^{n-2} + \dots + a_1 z + 1\text{,}\) and choose as \(\gg\) a circle that is large enough to make the condition of Rouché’s theorem work. You might want to first apply Proposition 5.3.1 to \(g(z)\text{.}\)
22.
Suppose \(S\subset\C\) is closed and bounded and all points of \(S\) are isolated points of \(S\text{.}\) Show that \(S\) is finite, as follows:
(a)
For each \(z\in S\) we can choose \(\phi(z)>0\) so that \(D[z,\phi(z)]\) contains no points of \(S\) except \(z\text{.}\) Show that \(\phi\) is continuous.
Hint.
This is really easy if you use the first definition of continuity in Section 2.1.
(b)
Assume \(S\) is non-empty. By the Extreme Value Theorem A.0.1, \(\phi\) has a minimum value, \(r_0>0\text{.}\) Let \(r=r_0/2\text{.}\) Since \(S\) is bounded, it lies in a disk \(D[0,M]\) for some \(M\text{.}\) Show that the small disks \(D[z,r]\text{,}\) for \(z\in S\text{,}\) are disjoint and lie in \(D[0,M+r]\text{.}\)
(c)
Find a bound on the number of such small disks.
